Mark Adams
Division of Psychiatry
mark.adams@ed.ac.uk
Genetics and Environmental Influences on Behaviour and Mental Health
Common: Affects 1% or more of the population
Complex: Inheritance cannot be explained by a single gene
Why use genetics to study mental health and psychiatric disorders?
Diagram showing the seven “characters” observed by Mendel
Adding up effects from a large number of genetic effects to make a continuous phenotype is related to the Central Limit Theorem.
Proportion of similarity in phenotypes that can be attributed to similarity in genotypes.
Model: Phenotype (P) = Genotype (G) + Environment (E)
Variance decomposition \[var(P) = var(𝐺) + var(𝐸)\] Proportion of variance \[h^2 = \frac{var(𝐺)}{var(𝑃)}\]
Plot of child (offspring) height versus the average of their parents’ heights. What is a statistic that can be used to summarise the relationship between these two variables?
\(\beta = \frac{\mathrm{cov}(A, B)}{\mathrm{var}(A)}\)
Estimate the beta coefficient (slope) for a simple regression from the covariance between predictor (\(A\)) and outcome (\(B\)) variable divided by the variance of the predictor (\(A\)).
\[ P = G + E \]
The phenotype value \(P\) is influenced by a genetic effect \(G\) and and environmental effect \(E\).
\[ G = d + s \]
Each individual has two copies of the genome, one inherited from each parent.
Phenotype (\(P\)) value is the sum of the two genetic values plus an environmental value (\(e\)).
\(\beta = \frac{\mathrm{cov}(A, B)}{\mathrm{var}(A)}\)
Therefore, \(\beta = \frac{\mathrm{cov}(\frac{P_d + P_s}{2}, P_o)}{\mathrm{var}(\frac{P_d + P_s}{2})}\)
\[ \mathrm{cov}(\frac{P_d + P_s}{2}, P_o) \]
\[ = \mathrm{cov}(\frac{d + d^\prime + e_d + s + s^\prime + e_s}{2}, d + s + e_o) \]
Expand the terms. Recall that:
\[ \mathrm{cov}(A+X,B+Y) = \\ \mathrm{cov}(A,B) + \mathrm{cov}(A,Y) + \mathrm{cov}(X,B) + \mathrm{cov}(X,Y) \] Thus we can do a pairwise expansion to: \[ = \mathrm{cov}(\frac{d}{2} + \frac{d^\prime}{2} + \frac{e_d}{2} + \frac{s}{2} + \frac{s^\prime}{2} + \frac{e_s}{2}, d + s + e_o) \] \[ = \mathrm{cov}(\frac{d}{2}, d) + \mathrm{cov}(\frac{d^\prime}{2}, d) + \dotsm+ \mathrm{cov}(\frac{e_s}{2}, e_o)\]
Covariance between a genetic effect and itself \[ \mathrm{cov}(\frac{d}{2}, d), \mathrm{cov}(\frac{s}{2}, s) \]
Covariance between genetic effects from the same parent \[ \mathrm{cov}(\frac{d^\prime}{2}, d), \mathrm{cov}(\frac{s^\prime}{2}, s) \]
Covariance between genetic effects from different parents \[ \mathrm{cov}(\frac{d^\prime}{2}, s), \mathrm{cov}(\frac{s^\prime}{2}, d) \]
Covariance between parent and offspring environment effects \[ \mathrm{cov}(\frac{e_d}{2}, e_o), \mathrm{cov}(\frac{e_s}{2}, e_o) \]
Covariance between parental genetic and offspring environmental \[ \mathrm{cov}(\frac{d}{2}, e_o), \mathrm{cov}(\frac{s}{2}, e_o) \]